Flow and Heat Transfer in Rotating Compressor Cavities With Inverted Shroud-Throughflow Temperature Differences

Figure 1 shows the primary gas path and secondary air system (SAS) in an aero-engine. The former flows directly from the compressor to the combustion chamber through a mainstream annulus. The SAS diverts compressed air to bypass the combustion process for auxiliary systems and turbine cooling. Compressor blades in the mainstream annulus are attached to co-rotating discs that form cavities exposed to this secondary flow, typically an axial throughflow at low radius. The discs feature cobs at low radius and a shroud through which heat is conducted from the mainstream.

The interaction of the axial throughflow (in the annular gap between the disc cobs and the shaft) and the rotating cavity creates a conjugate problem. In the high-pressure (HP) compressor, the shroud is heated by compression of air in the main gas path, while the cobs are cooled by the secondary flow. The radial temperature gradient across the rotor discs is coupled to the difference in temperature between the disc surface and air, resulting in the formation of buoyancy-induced flow. The flow can be unstable and typically takes the form of counter-rotating vortex pairs separated by radial hot and cold plumes. This unsteady flow structure in the cavity affects the heat transfer to the discs, which in turn affects the disc temperature gradient in a conjugate process. Accurate predictions of disc temperatures are crucial as these affect thermal stresses, disc growth, and hence changes in compressor blade tip clearance throughout an engine cycle. Thermal stresses also govern the operating life of the discs.

Figure 1 also illustrates the SAS for intermediate pressure (IP) compressors, where the secondary air is shown to be directed upstream (forwards relative to the main gas path). In this inverse scenario, the axial throughflow is hotter than the compressor discs, reversing the disc temperature gradient with the cob hotter than the shroud. This forward-flow configuration may be used to ensure a sufficiently high secondary air pressure, with the flow sourced from later stages of the compressor. This technology is further described by Kroes and Wild [1]. Additionally, secondary air directed upstream can be used in the fan de-icing system [2]. Due to the nature of the configuration, cavities exposed to the inverted temperature gradient can operate in this regime constantly, or experience temperature gradient inversion during transient operation. Inverted temperature gradients can typically occur in the HP compressor during engine transients in the flight cycle.

Despite the practical application and importance, the inverted temperature gradient configuration has not been previously investigated. This work presents an experimental study using the University of Bath Compressor Cavity Rig, investigating this phenomenon for the first time. The experiments were conducted over a range of engine-representative Rossby numbers. For context, experiments were also performed using the conventional cooled throughflow scenario as well as isothermal conditions.

Owen and Long [3] show that the fluid dynamics and heat transfer in compressor cavities are governed by the following non-dimensional parameters (see Nomenclature): Nusselt (⁠$Nu$⁠), Rossby (⁠$Ro$⁠), Grashof (⁠$Gr$⁠), and rotational Reynolds (⁠$Reϕ$⁠) numbers, and the buoyancy parameter $βΔT$⁠. Such flows are dominated by buoyancy-induced convection with $Gr∼1013$⁠. The flow is three-dimensional, unsteady, and unstable, with a self-organizing system causing great uncertainty in the design process. Circumferential Coriolis forces are created by the pressure distribution from cyclonic and anticyclonic vortex pairs in the core and shear stresses in the Ekman layers near the discs. In the HP compressor with a heated shroud and positive temperature gradient (⁠$βΔT>0$⁠), cold air flows radially outward and hot air flows radially inward as illustrated in Fig. 2(a). The flow structure is influenced by heat and fluid exchange with the axial throughflow. The cavity can be partitioned into an inner region dominated by ingress and egress from the throughflow, and an outer region dominated by buoyancy.

The flow structure for the inverted temperature gradient (⁠$βΔT<0$⁠) is shown in Fig. 2(b). Here the axial throughflow is hotter than the shroud and discs, and the radial motion of the hot and cold plumes is reversed. Experimental results presented in Sec. 4 will show that symmetric (similar-sized) cyclonic and anticyclonic vortices form in this scenario. This differs from the case with $βΔT>0$⁠, where there is an asymmetry in the sizes of the vortex pairs. The counter-rotating flow structures are influenced by buoyancy due to heat transfer with the disc cobs.

This section reviews the effects of the axial throughflow on the cavity flow structure and heat transfer. The degree of interaction depends principally on the geometry and the $Ro$⁠. In aero-engines (Fig. 1) the cavities are typically open, with the thickness of the disc cobs governed primarily by stress considerations. An alternative (canonical) case is a closed cavity, where there is no interaction with the throughflow. A near-closed cavity configuration with thick cobs is commonly found in industrial gas turbines. Nicholas et al. [4] investigated the effects of variable cob geometry on disc heat transfer and flow structure in the cavity.

Owen and Pincombe [5] and Farthing et al. [6] conducted early smoke visualization and laser Doppler anemometry (LDA) experiments in an isothermal scenario with $Ro>1$⁠, observing that the extent of flow penetration into the cavity changed with $Ro$⁠. A toroidal vortex was observed to form at the cavity entrance and increased in strength with increasing $Ro$⁠. Andereck et al. [7] investigated the Taylor–Couette flow features between rotating concentric cylinders and observed a range of possible flow states.

Jackson et al. [8–10] discussed steady-state experiments conducted in the Bath Compressor Cavity Rig using an open cavity configuration, showing that disc Nusselt numbers increased with increasing $Ro$⁠. The slip (relative to the disc) of the asymmetric vortex structures was affected by $Ro$⁠, with peak slip occurring at $Ro=0.4$⁠. Flow reversal was identified in the axial throughflow for $Ro<0.4$⁠. As illustrated in Fig. 3, subcritical (⁠$Ro<0.3$ for this research) and supercritical flow regimes (⁠$Ro>0.3$⁠) may be defined: in the former, reversal flow is prominent; in the latter, the toroidal vortex in the cob region is dominant. Many of these experimental findings were later captured by wall-modeled large eddy simulation (WMLES) [11], with increased unsteadiness and fragmentation of vortices and radial plumes at $Ro=0.2$⁠. The flow reversal was also shown to strengthen at lower $Ro$⁠. Gao and Chew [12] investigated higher Rossby numbers (⁠$0.4−0.8$⁠) using WMLES. While significant mass flow exchange was found at high rotational speed and $Ro=0.4$ (42.6% of the bore flow), the mass exchange as a fraction of bore flow reduced with increasing $Ro$⁠. Similar effects were noted by Long [13] and Günther et al. [14].

Atkins and Kanjirakkad [15] presented an experimental and hybrid RANS/LES study of the Sussex Multiple Cavity Rig for $0.3<Ro<4.74$⁠. Unsteadiness of the toroidal vortex was noted at high $Ro$⁠. This toroidal vortex was suppressed with increasing $Reϕ$ around $Ro=1$⁠. Cases at $Ro=0.3−0.6$ showed sensitivity to changes in buoyancy effects. LDA measurements from Fazeli et al. [16] highlighted the existence of multiple flow regions in the cavity. Near solid body rotation was found in the outer region, while interaction with the bore flow was demonstrated near the inner region. Fisher and Puttock-Brown [17] experimentally investigated the toroidal vortex in the same rig, using a miniature five-hole probe in a heated shroud configuration at $0.11<Ro<3.24$⁠. The toroidal vortex was shown to be asymmetric and biased toward the downstream disc, consistent with previous studies [11,15,18]. The vortex strength and size increased with increasing $Ro$ between 0.34 and 1.63. The effect of $βΔT$ was found to be negligible compared to the effect of $Reϕ$ with the Coriolis force suppressing local circulation at higher $Reϕ$⁠. Radial exchange between the cavity air and the throughflow reduced with increasing $Ro$⁠, consistent with other research [12–14].

Saini and Sandberg [18] performed LES simulations at $Ro=4.5$⁠, based on the experimental conditions from Ref. [15]. A toroidal vortex of large radial extent was observed, with a rise in the inner region temperature and a reduction in density attributed to significant flow mixing. Negative heat transfer from the air to the discs was computed in the toroidal vortex region. It was noted that the bore flow did not impact the heat transfer near the shroud, consistent with the findings of Puttock-Brown et al. [19] and Pitz et al. [20].

Hickling and He [21] conducted conjugate heat transfer computations on the experimental rig of Bohn et al. [22] with two heating configurations—axially heated discs with an adiabatic shroud, and a radially heated shroud with adiabatic external disc boundary conditions. It was observed that the cavity flow structure was strongly dependent on the heating configuration. Strong shroud heat transfer in the latter configuration caused the formation of buoyancy-induced vortex streaks on the shroud that enhanced fluid mixing, consistent with the findings of Puttock-Brown et al. [19]. In the former configuration, the vortices formed in the radial outflow with significant near-disc instabilities caused by disc heat transfer (near enough to affect the Ekman layers). The authors noted that heating the air near the disc surface reduces local density, leading to a reduction in the density-dependent Coriolis force, which has been previously observed to have a stabilizing effect on the flow structure (notably in direct numerical simulation of Pitz and Wolf [23,24]). The average disc Nusselt number was higher in the axially-heated configuration. The stabilizing effect of rotation was also found in simulations of Gao and Chew [12], who observed a constant rothalpy core forming in the cavity at high rotational speed, appearing to reduce the driving buoyancy effects.

An overview of different numerical models and their applicability to an open cavity case was examined by Hickling and He [25]. A significant influence of the disc heating boundary conditions on the predicted cavity flow structure was shown. The authors highlighted the importance of conjugate heat transfer simulations to accurately capture the unsteady flow structures. A steady–unsteady coupling conjugate model for a closed cavity was introduced by Parry et al. [26]. Wang et al. [27] compared numerical models on a wide range of geometries and demonstrated the applicability of open-source computational fluid dynamics (CFD) software.

Using the Bath Compressor Cavity Rig, Lock et al. [28] investigated flow stratification and compressibility effects in the cavity core as $βΔT→0$ in a closed cavity. Pernak et al. [29] experimentally assessed heat transfer in transient conditions. Nicholas et al. [30] presented a new predictive theoretical model of heat transfer and flow structure in an open compressor cavity, as well as a transient model for a closed cavity [31]. Both models were validated using experimental data.

The vast majority of published research investigating rotating compressor cavities has considered positive gradients in temperature, which dominate in the HP compressor. Isothermal cavities have also been studied. Despite the practical importance in terms of engine operation and design, to date, there are no published studies investigating the effects of heating the axial throughflow in conjunction with a cooled shroud. This work presents an experimental study using the University of Bath Compressor Cavity Rig, investigating the consequences of an inverted temperature gradient for the first time. Any previous discussion in the literature has been speculative. Notably, Owen and Long [3] expected thermal stratification and minimal heat transfer from the shroud and discs.

The Bath Compressor Cavity Rig is shown in Fig. 4. It was designed to simulate a generic axial compressor at fluid-dynamically scaled conditions. A detailed description of the facility, operating range and instrumentation is provided by Luberti et al. [32]. This section discusses the design modifications made to conduct the inverted temperature gradient (heated throughflow) experiments.

The core of the rig is a rotating drum formed of four titanium discs overhung on a shaft. There are three inter-disc cavities, with the two outer cavities lined with Rohacell (low-conductivity insulating foam) to create quasi-adiabatic boundary conditions for the central cavity. Rohacell inserts isolate the entrances to the upstream and downstream cavities. The instrumented central cavity forms the test section, which is open to the axial throughflow. An internal stationary shaft of 52 mm radius supports additional instrumentation.

Figure 5 shows the instrumented disc (⁠$a/b=0.29$⁠) and the inner annulus. Also shown is the stationary shaft with a thermocouple rake (three in total). The shroud and cob radii are $b=240mm$ and $a=70mm$⁠, respectively. The constant-thickness disc diaphragm extends between radii 124 mm and 235 mm. The radial distribution of temperature was measured using an array of K-type thermocouples in four radial arms. The thermocouples were embedded in circumferential grooves spanning an isotherm at a constant radius, using epoxy resin with thermal properties closely matching titanium to minimize thermal disturbance error. An RdF 27160-C-L-A01 thermopile heat flux gauge was mounted onto the shroud of the downstream disc; its calibration is described by Pountney et al. [33]. Two fast-response Kulite XCQ-080 sensors monitor the pressure variation within the cavity. The sensors are mounted flush with the surface of the downstream disc at $r/b=0.85$ and separated circumferentially by $35deg$⁠. The surfaces of the discs and stationary shaft were painted matt black to allow for an accurate calculation of radiation [34]. The opening to the cavity can be modified using aluminum rings attached to the cobs [4], but here the cavity opening was at its maximum with $s′/s=0.65$⁠, $G=s/b=0.11$⁠.

In addition to the rotating instrumentation, thermocouple rakes were mounted on the stationary shaft to measure the throughflow temperature distribution upstream and downstream of the test section. The rakes typically contained five evenly-spaced K-type thermocouples with a slim cylindrical section to minimize flow disruption.

Wires from the rotating instrumentation were passed through the disc drum and rotating shaft to the drive end. A Datatel telemetry unit and antenna transmitted data to a stationary unit. The signals were converted to temperature using cold-junction measurements from PT100 resistance temperature detectors and a known calibration. The sampling rate used to collect temperature and heat flux data was 1 Hz. Pressure data were collected at 10 kHz via a National Instruments NI 9215 data logger and filtered using a 1 kHz low-pass filter. The maximum rotational frequency of the disc was 133 Hz (8000 rpm), so the acquisition and filtering frequencies were sufficiently high to satisfy the Nyquist criterion. Estimated measurement uncertainties are as follows: $±0.5∘C$ for rotating and $±0.3∘C$ for stationary thermocouples respectively, $±2.5−4%$ for the heat flux gauge, and $±1.5mbar$ for the pressure sensors.

A 30 kW electric motor spun the assembly via a belt drive, with a motor encoder controlling the speed to $±10rpm$⁠. Ambient air was drawn from the laboratory by a pump and passed through the annulus between the stationary shaft and cobs of the rotating discs. The pump delivered a flowrate of up to $0.15kg/s$⁠, monitored by a thermal mass flowmeter with $±5×10−4kg/s$ accuracy. A conventional bell-mouth inlet was replaced with a volute, 3D-printed from PC-ABS plastic, to facilitate heating this flow. As shown in Fig. 4(b), the volute provided a convenient way of supplying the heated throughflow. The flow transitioned from a circular air intake to the axial throughflow annulus. The volute geometry was designed using CFD simulations to ensure a uniform velocity distribution in the annulus. A labyrinth seal closed the gap between the stationary and rotating parts, and heat loss was minimized by lining the outer surface of the annulus with Rohacell foam. The thermocouple rakes were used to ensure the temperature profile in the axial throughflow was appropriate for the experiments.

Two heater assemblies were used to create positive and inverse temperature gradients. A circular array of six (2 kW) radiant heater filaments surrounded the disc pack, heating the shroud with a cool axial throughflow. To control windage heating, the radiant heater assembly was mounted on rails and connected to a lead screw. The temperature of the shroud was limited to $100∘C$⁠, resulting in a maximum $βΔT=0.25$⁠. The assembly was moved to expose the shroud to relatively cool ambient air for negative $βΔT$ tests, with an additional heater (Secomak model 571) connected to the volute. This 3 kW heater was operated using a proportional–integral–derivative (PID) controller to maintain the temperature of the axial throughflow with $2∘C$ accuracy. This air temperature was limited to $130∘C$⁠, achieving a minimum $βΔT=−0.15$ for the experiments with an inverted temperature gradient. All experiments presented here were conducted using the same inlet configuration (volute with throughflow heater attached) to eliminate any effects of changing geometry. The radial distributions of temperature on the discs with the volute were very similar to those with the axisymmetric bell-mouth inlet. The spectrograms presented in Sec. 4 across all Rossby numbers under isothermal conditions revealed qualitatively similar structures with the volute and with the bell-mouth.

The distribution of $τd$ is found by optimizing the agreement between the calculated and experimentally measured values of $θd$ while satisfying the smoothing conditions using the Bayesian approach and a probabilistic interpretation. The method provides the heat fluxes from the discs, a statistical curve of resultant Bayesian temperatures and the corresponding 95% confidence interval. As these values are based upon experimental measurements, the resultant radial distributions of heat flux and temperature are referred to as experimentally derived. All heat flux values presented in this paper were corrected for radiation using the method described in Ref. [34].

In this section, temperature, heat flux, and unsteady pressure data are presented for heated axial throughflow in both the subcritical and supercritical Ro regimes. The measurements are compared to those under positive temperature differences (heated shroud) and isothermal conditions. All data were recorded under steady-state conditions where the disc temperature variation within 10 min was less than 1K.

Figure 6(a) shows the radial distribution of disc temperature for four cases at subcritical $Ro$⁠: an inverted temperature gradient where $βΔT=−0.12$⁠, two positive temperature gradients at $βΔT=+0.12$ and $+0.25$⁠, and an isothermal case with $βΔT=0.00$⁠. Note that for the latter, the flow is not technically isothermal but the dynamic temperature rise was $<2K$⁠. All cases were conducted at 2000 rpm and $Ro=0.2$⁠. The axial throughflow temperatures were measured from the rakes upstream and downstream of the cavity and are shown for three cases for $0.05<r<0.07$⁠. Also shown are Bayesian curves, along with the 95% confidence intervals. The insert shows further detail. Figure 6(b) shows the disc heat flux using the Bayesian model, which is only available across the diaphragm of the disc. The silhouette showing the geometry of the cavity is aligned with the vertical axis.

For $βΔT=0.00$⁠, the variation of disc temperature was within 2 K and the resultant disc heat flux is close to zero, $|τd|<0.5W/m2$⁠, showing a virtual isothermal condition. For positive $βΔT$⁠, disc temperatures were higher than the throughflow temperature and increased with increasing radius. The disc heat fluxes were generally positive with the magnitude increasing with radius, revealing heat transfer from the warm disc to the cold fluid core in the cavity (⁠$Td>Tc$⁠). Heat and mass exchange with the cavity causes an increase in throughflow temperature. For clarity, this change in temperature is marked in Fig. 6 only for $βΔT=+0.25$ (not $βΔT=+0.12$⁠).

The discs were cooler than the throughflow for the case where $βΔT=−0.12$⁠. The radial variation of temperature is qualitatively similar to that at $βΔT=+0.12$ but in an opposite (inverted) sense; the heat flux is of a smaller magnitude and entirely negative, meaning convective heat transfer to the disc with $Td<Tc$⁠. The disc heat flux approached zero at high radii, indicating suppressed heat transfer and negligible penetration of the throughflow to the shroud. The heat transfer is greatest at low radii, which has led to an unexpected phenomenon. Rather than the predicted thermal-stratification within the cavity, there is buoyancy-induced flow (and associated rotating structures, which were not observed at $βΔT=0.00$⁠) attributed to the temperature difference between the warm fluid core and the cooler cobs. There is a slight reduction in axial throughflow temperature downstream of the cavity due to an inverted enthalpy exchange.

The non-dimensional forms of the radial variation of temperature and heat flux are presented in Fig. 7 (with the isothermal case removed). These cases are all at common Rossby and rotational Reynolds numbers. The remarkable differences in the magnitude of temperature gradients and heat fluxes suggest that $βΔT$ significantly influences the non-dimensional heat flux.

Figure 8 illustrates the effect of $Reϕ$ for two cases with an inverted temperature gradient, at common $βΔT$ and $Ro$⁠. The magnitude of negative heat flux from the disc at low radii increases as $Reϕ$ increases. This enhanced heat transfer is consistent with previous work for cases with positive $βΔT$ [37]. In both cases, the heat flux approaches zero toward the shroud. The disc temperature is increased with the increased forced convection between the inner cob surface and axial throughflow.

Entrainment of throughflow is dominated by the Rossby number, which (in turn) affects the heat transfer within the cavity. Figure 9 shows the effect of varying $Ro$ on disc temperature and heat flux for three cases at $βΔT=−0.12$⁠. As $Ro$ is decreased from 0.20 to 0.11, there is a clear decrease in disc and cob temperatures. There is also a qualitative shift in the shape of the radial distribution of temperature, moving toward the solution for pure conduction in an annular fin [28]. There is a significant decrease in the magnitude of heat flux as $Ro$ decreases due to reduced entrainment of hot throughflow into the cavity. Where the radial distribution of temperature approaches the conduction solution, the convective heat flux is negligible. At high $Ro$⁠, there is significant disc heat flux near the shroud, suggesting the hot entrained air penetrates to high radius.

The unsteady flow structure was also examined for all cases. Remarkably, experimental evidence shows that vortical flow structures form in the rotating cavity under conditions of heated throughflow (negative $βΔT$⁠). Figure 10 shows the FFT plots of the pressure signals collected in the frame of reference of the rotating disc at $Ro=0.2$ and $βΔT=−0.12$⁠, 0.00 and $+0.25$⁠. For the isothermal cavity, there is no dominant frequency, indicating there were no prominent structures at subcritical Ro. A primary peak was observed for $βΔT=+0.25$⁠, showing a principal structure frequency (slip) at approximately 15%. The secondary peak at $f/fd∼30%$ indicates asymmetry of the vortex pair, as was evidenced in Ref. [10]. An explanation of how the peaks on the FFT plot are interpreted as indicators of vortex asymmetry is shown in the  Appendix. A cross-correlation of the pressure signals from the two sensors reveals the typical buoyancy-induced flow structure in an open rotating cavity: a single vortex pair, where the anti-cyclonic vortex is smaller in size.

A well-defined peak in the FFT was also observed for the negative $βΔT$ case. The slip of the structures was higher at approximately 22%. Cross-correlation of the two pressure signals shows only one pair of vortices. However, the secondary peak in the FFT was not clearly defined, indicating that the asymmetry of the structures was reduced. The circumferential pressure distribution against the structure revolutions is shown in Fig. 11 for positive and negative $βΔT$⁠. Regions of positive and negative pressure coefficient are attributed to the anticyclonic and cyclonic vortices, respectively. While vortex asymmetry is reduced at negative $βΔT$⁠, the relative size of vortices has changed—here the anti-cyclonic vortex is marginally larger in size. The broad flow structures for both heated and cooled axial throughflow are further illustrated in Fig. 2.

In the subcritical Ro regime, vortical flow structures were formed in the rotating cavity due to the effects of buoyancy. While the effect of buoyancy at positive $βΔT$ is well understood, the effect under negative $βΔT$ (heated throughflow) was unexpected; it was previously believed that the flow would be stratified under such conditions. However, the disc temperature and heat flux distributions shown in Fig. 8 indicate the cavity air is hotter than the disc at low radius, leading to local buoyancy-induced structures within the cavity. The radial plumes of hot and cold air will be at the maximum gradient of pressure coefficient in Fig. 11. The magnitude of the plume mass flowrate is proportional to the circumferential pressure difference [37], and hence less for $βΔT=−0.12$ relative to $βΔT=+0.25$⁠. The sizes of anticyclonic and cyclonic vortices are the combined result of momentum balance in the cavity and these plume mass flowrates. Generally, larger sized anti-cyclonic vortices lead to higher structure slip. It is also possible that the vortex core shifts radially with a change in $βΔT$⁠.

Figure 12 presents the non-dimensional disc temperature and heat flux distributions for varying $βΔT$ at a supercritical Rossby of $Ro=0.4$⁠. There is qualitative similarity and symmetry for the temperature profiles with negative and positive $βΔT$ at the same $Ro$⁠. The heat flux for the case of inverted $βΔT$ is again negative. In contrast to the subcritical case at $Ro=0.2$ (with near zero heat flux near the shroud), the magnitude of the disc heat flux increases with increasing radius. This increased heat transfer reveals significant penetration of heated throughflow to high radius in the cavity.

The higher-intensity flow structure in the cavity is also revealed by pressure measurements. Figure 13 presents the FFT of the unsteady pressure signals collected at $Ro=0.4$ and $βΔT=−0.12,0.00,+0.12$⁠, and $+0.25$⁠. For all cases the peak $Cp$ are higher than those at $Ro=0.2$⁠, showing stronger vortical structures. Importantly, flow structures are observed at isothermal conditions, in stark contrast to the measurements in the subcritical regime. There is a clear principal peak in $Cp$ with a slip of approximately $14%$ for $βΔT=0.00$⁠. All peak frequencies and peak magnitudes of $Cp$ in Fig. 13 are similar, despite the wide range of $βΔT$⁠. The interpretation is that the flow structure at supercritical Rossby numbers is not primarily driven by buoyancy effects, but instead is strongly affected by the throughflow and formation of the toroidal vortex at low radius. This is examined further in Sec. 5.

A cross-correlation of pressure signals from the two sensors confirmed the presence of a single vortex pair for all cases. Introducing heating results in an increase in slip for both positive and negative $βΔT$⁠. This is consistent with the relative sizes of the cyclonic and anticyclonic vortices. The asymmetry is most pronounced for the isothermal case, where the anticyclonic vortex (shown by the positive pressure) is smaller than the cyclonic vortex; for positive $βΔT$⁠, the asymmetry is present but reduced in magnitude. At negative $βΔT$ the anti-cyclonic vortex is marginally larger (see also Fig. 2). As discussed in Sec. 4.1, the asymmetry is caused by the combined effects of momentum balance and plume mass flowrates. The circumferential pressure difference in Fig. 14 indicates that the plume mass flowrate for the isothermal case is higher than that of the negative $βΔT$ case and lower than the positive ones, contributing to variation in the slip speed and vortex asymmetry.

Figure 15 illustrates the effect of Ro on non-dimensional temperature and heat flux for $βΔT=−0.12$ in the supercritical regime. The distributions of $θ$ and $τd$ are consistent for both supercritical cases, indicating a similar flow structure within the cavity. Here there is significant ingestion of hot axial throughflow with non-zero heat flux at high radius.

The Rossby number is an important governing parameter for the flow structure in the cavity. This section describes experiments conducted over the range $0<Ro<0.6$ for cases with negative, zero, and positive $βΔT$⁠. These experiments operated at a constant rotational speed (i.e., constant $Reϕ$⁠) but with continuously varying axial throughflow mass flowrate over 30 min. The throughflow was increased and decreased to ensure no effect of hysteresis. Pernak et al. [29] showed that the flow structure in the cavity reacted virtually instantaneously to a change in $Ro$⁠, despite a longer time constant for the disc temperatures due to thermal inertia. Further confirmation was made by comparing the structure activity for the variable $Ro$ experiments with those conducted under steady-state conditions.

Figure 16 shows a spectrogram from the unsteady pressure transducer as a function of $Ro$ for isothermal conditions (⁠$βΔT=0$⁠). High values of $Cp$ denote peaks in pressure coefficient, with the ordinate showing the peak frequency of the flow structures (if any) as a fraction of the rotational speed of the disc. The abscissa is the Rossby number varying continuously with time in a quasi-steady manner. Flow structures are clearly present in the cavity for $Ro>0.3$ in the supercritical regime, where the flow was dominated by a toroidal vortex. Increased unsteadiness was detected for $Ro>0.21$ in the form of a continuum as the intensity of the vortex increased with increasing $Ro$⁠. There is a variation of between 15% and 20% for the slip of the structures. The secondary peak at $f/fd>0.3$ is related to the asymmetry of the single vortex pair. The critical Rossby number separating the subcritical and supercritical regimes was approximately 0.3. This value would be dependent on the swirl of the axial throughflow and the geometry of the cavity. For subcritical$Ro$⁠, the flow structure in the cavity would be dominated by buoyant heat transfer and no structures were observed here under isothermal conditions.

Figure 17 repeats the spectrogram for the isothermal case, with data added for $βΔT=+0.25$ (cool throughflow and heated shroud) and $βΔT=−0.12$ (heated throughflow and shroud at cooler temperature). For clarity, the scale is restricted to $f/fd<0.25$ but all spectrograms featured a single vortex pair in the cavity. The Rossby number was again varied continuously over 30 min with an imperfect thermal quasi-steady-state, limited by the thermal inertia of the rig. Variation in $Ro$ inevitably affected the temperature distribution across the discs but the changes in $βΔT$ during the experiment were less than 10% over the full range of $Ro$⁠. Again, steady-state tests were conducted at the chosen $Ro$ to confirm observations from the continuous sweep.

Figure 17 shows that structures form in the cavity below the critical $Ro$ for both positive and negative $βΔT$⁠. This is a result of the radial temperature gradient in the fluid core driving buoyancy effects and the formation of cyclonic and anticyclonic vortices. For $βΔT=+0.25$⁠, the slip increases with $Ro$ up to the critical value. In the supercritical regime, the structures are dominated by the axial throughflow and toroidal vortex, and the slip stabilizes at about 17%. At $Ro$ near the critical value, there is a combined effect between the buoyancy forces and the toroidal vortex, with the slip peaking at 20%. Consider the case with an inverted temperature gradient for negative $βΔT=−0.12$⁠. The qualitative behavior of the cavity flow structures is similar to that with positive $βΔT$ but reversed. In the subcritical regime, the effect of buoyancy at positive $βΔT$ is well understood. The effect under negative $βΔT$ can be explained by the disc temperature and heat flux distributions shown in Fig. 9. These indicate the cavity air is hotter than the disc at low radius, leading to local buoyancy-induced structures within the cavity. There is a notably higher measured slip in the subcritical regime with negative $βΔT$⁠, which can be explained by the lower buoyancy-driven plume mass flowrate at low Ro.

The shroud heat flux was also affected by the Rossby number. Figure 18 presents the measured variation of non-dimensional shroud heat flux with $Ro$ for both positive and inverted temperature gradients. The data for $βΔT=+0.25$ show the heat flux increases with increasing $Ro$ due to increased mass and heat exchange between the cavity and the bore flow. Peak heat transfer can be observed around the critical $Ro$ where the toroidal vortex forms at the cavity entrance. In the supercritical regime, the heat flux stabilizes as the flow is dominated by the toroidal vortex, consistent with the stabilizing structure slip shown in Fig. 17. The situation is different for the inverted temperature gradient case. In the subcritical regime, the shroud heat flux is effectively zero (within measurement uncertainty). This is due to the limited entrainment of hot fluid to high radius in the cavity. At supercritical $Ro$⁠, the toroidal vortex encourages entrainment of the axial throughflow and the (negative) shroud heat flux increases as hot air reaches the relatively cool shroud. This is consistent with changes in disc heat flux distribution presented in Figs. 9 and 15. Note that the shroud heat flux in the negative $βΔT$ case is an order of magnitude lower than that observed with positive $βΔT$⁠.

This marks the first study of flow and heat transfer in a rotating compressor cavity with heated axial throughflow. The inverted temperature gradient occurs in the practical working conditions of aero-engine LP and IP compressors. The University of Bath Compressor Cavity Rig has been specifically designed to simulate such flows, measuring disc temperature and unsteady pressure in the frame of reference of the rotating discs. Bayesian and spectral analysis were used to determine the radial distribution of disc heat flux and the unsteady nature of vortex structures in the rotating core of the cavity. Experimental measurements were acquired over a range of engine-representative Rossby number (⁠$Ro$⁠) and buoyancy parameters (⁠$βΔT$⁠). The study included positive temperature gradients (⁠$βΔT>0$⁠) that occur in the HP compressor, where the flow and heat transfer are dominated by buoyancy-induced dynamics. For context, isothermal conditions (⁠$βΔT=0$⁠) were also investigated.

Of principal importance, the flow structures in cavities with positive and inverted (⁠$βΔT<0$⁠) temperature differences were unexpectedly shown to be fundamentally similar (albeit with reversed radial temperature profiles). Isothermal cases identified a critical $Ro$⁠, above which the flow structure was dominated by a toroidal vortex. At subcritical $Ro$⁠, the flow structure for the inverted temperature gradient was governed by buoyancy due to disc heat transfer and the ingestion of heated throughflow to high radius in the cavity. The radial distribution of non-dimensional temperature was shown to be qualitatively similar over a wide range of positive and negative $βΔT$ at supercritical $Ro$⁠. Increasing $Ro$ increased the heat flux across the discs. At low $Ro$⁠, there was limited heat and fluid exchange between the cavity and axial throughflow; for cases with negative $βΔT$ the convective heat flux from the discs approached zero and the radial distribution of temperature matched the solution for pure conduction in an annular fin.

The cavity flow structure was significantly affected by $Ro$⁠. Cross-correlation of pressure signals identified a single pair of cyclonic and anticyclonic vortices in all cases. The asymmetry of these rotating structures and slip relative to the disc was also shown to depend on $βΔT$⁠. Introducing heating increased the slip for both positive and negative $βΔT$⁠, consistent with the relative sizes of the cyclonic and anticyclonic vortices. The asymmetry was most pronounced for the isothermal case, where the anticyclonic vortex was smaller than the cyclonic vortex; for positive $βΔT$⁠, the asymmetry was present but reduced in magnitude. At negative $βΔT$ the anti-cyclonic vortex was marginally larger. This asymmetry was caused by the combined effects of momentum balance and radial movement of hot and cold plumes through the rotating core of the cavity.

This research provides fundamental insight into the complex phenomenon of buoyancy-induced rotating flow. The results are also of practical importance to the engine designer dealing with disc cavities in LP and IP compressors and inform the development of accurate, physics-based, thermo-mechanical design codes.

The research presented in this paper was supported by the UK Engineering and Physical Sciences Research Council in collaboration with Rolls-Royce plc and the University of Surrey, under the Grant No. EP/P003702/1. The authors are especially grateful for the support of Jake Williams and the approval from Rolls-Royce plc to publish the work.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$a$ =

inner radius of the cavity (cob) (m)

$b$ =

outer radius of the cavity (shroud) (m)

$c$ =

disc thickness at the diaphragm (m)

$h$ =

heat transfer coefficient (⁠$W/m2K$⁠)

$k$ =

thermal conductivity (⁠$W/mK$⁠)

$n$ =

number of vortex pairs

$p$ =

static pressure (Pa)

$q$ =

heat flux (⁠$W/m2$⁠)

$r$ =

radius (m)

$s$ =

axial cavity width (m)

$t$ =

time (s)

$x$ =

non-dimensional radial location

$G$ =

cavity gap ratio

$R$ =

gas constant (⁠$J/kgK$⁠)

$T$ =

temperature (K)

$W$ =

axial throughflow velocity (m/s)

$p¯$ =

mean static pressure (Pa)

$s′$ =

cob separation (m)

$fd$ =

rotational frequency of discs (Hz)

$fs$ =

rotational frequency of the flow structure (Hz)

 $fs,1$ =

rotational frequency of one vortex pair (Hz)

 $α$ =

angular separation of unsteady pressure sensors (rad)

$β$ =

volume expansion coefficient (⁠$K−1$⁠)

$γ$ =

ratio of specific heats

$θ$ =

non-dimensional temperature (⁠$=T−Tf|Tsh−Tf|$⁠)

$μ$ =

dynamic viscosity (⁠$m2/s$⁠)

$ρ$ =

density (⁠$kg/m3$⁠)

$τ$ =

non-dimensional heat flux (⁠$=b2qkdc|Tsh−Tf|$⁠)

$χ$ =

compressibility parameter (⁠$=Ω2b2γR(Tsh−Tf)$⁠)

$Ω$ =

angular velocity of disc (rad/s)

$d$ =

value on the disc surface

$f$ =

value in the axial flow

$sh$ =

shroud

$ϕ$⁠,$r$⁠,$z$ =

circumferential, radial, and axial directions

$Cp$ =

pressure coefficient (⁠$=p−p¯12ρfΩ2b2$⁠)

Gr =

Grashof number (⁠$=Reϕ2βΔT$⁠)

Nu =

Nusselt number (⁠$=qLkΔT$⁠)

$Reϕ$ =

rotational Reynolds number (⁠$=ρΩb2μf$⁠)

Ro =

Rossby number (⁠$=WΩa$⁠)

$βΔT$ =

buoyancy parameter (⁠$=Tsh−TfTf$⁠)

The cyclonic and anticyclonic vortices appear symmetric both in angular extent and magnitude of pressure. Figure 20 shows a fast Fourier transform of the measured unsteady pressure data, as well as an FFT of data generated with the fitted sine curve. A single clear peak at $f/fd=0.22$ is observed for both measured and fitted data, indicating a structure slip of 22%.

The anti-cyclonic (high pressure) vortex covers a larger angular extent than the cyclonic (low pressure) vortex, but with a lower magnitude of $Cp$⁠. Figure 22 shows the resultant frequency spectra for both the measured and fitted pressure signals. There is a clear second peak corresponding to $2×fs,1/fd$⁠. This shows that the primary peak in the frequency spectrum is linked to structure slip, as discussed earlier, and the secondary peak indicates asymmetry in the pressure distribution.